Mathematics puzzles

1. Aliens have abducted 100 human scientists to test their intellect. They prepared 101 robes numbered 1 to 101 with numbers painted behind their back of each robe. On the day of the test a random robe will be put aside and the rest of them will be randomly put on the scientists. Further the scientists will stand in line, also in random order, so that everyone sees the number of their colleagues in front of them but not his own and not of those who are behind him.Then, in any order they want they guess their own number. Each hears all previously made guesses but other than that they cannot speak. They cannot repeat a number that has already been guessed. The person who guesses wrong is killed on the spot. The ones who guess correctly will be allowed to go free. What strategy should the scientists agree upon in order to minimize the expected number of casualties? 
2. Distribute 127 one dollar bills among 7 wallets so that any integer sum from 1 through 127 dollars can be paid without opening the wallet. 
3. The digits 0-9(0,1,2,3,4,5,6,7,8,9) can be rearranged into 3628800 distinct 10 digits numbers.How many of these numbers are prime ? 
4. Consider the sum of the first 10 numbers: 1+2+3+4+5+6+7+8+9+10. Can you change some of the plus signs to minus signs so that the resulting sum is 0? 
5. Where Did the missing square go?
   
6. In a chessboard If we remove the two corner (diagonally opposite to each other) . Can we place 31 dominoes so that all the remaining 62 squares are covered. 
7. With one straight cut you can slice a pie into two pieces. A second cut that crosses the first one will produce four pieces, and a third cut can produce as many as seven pieces. What is the largest number of pieces that you can get with six straight cuts? 
8. You are given two eggs, and access to a 100-storey building. Both eggs are identical. The aim is to find out the highest floor from which an egg will not break when dropped out of a window from that floor. If an egg is dropped and does not break, it is undamaged and can be dropped again. However, once an egg is broken, that’s it for that egg.If an egg breaks when dropped from floor n, then it would also have broken from any floor above that. If an egg survives a fall, then it will survive any fall shorter than that.What strategy should you adopt to minimize the number egg drops it takes to find the solution?. (And what is the worst case for the number of drops it will take?) 
9. Probability of the Center of a Square Being Contained in A Triangle With Vertices on its Boundary.
10. Today is Sunday. Sleeping Beauty drinks a powerful sleep potion and falls asleep.
    Her attendant tosses a fair coin and records the result.
    The coin lands in Heads. Beauty is awakened only on Monday and interviewed. Her memory is erased and she is again put back to sleep.
    The coin lands in Tails. Beauty is awakened and interviewed on Monday. Her memory is erased and she's put back to sleep again. On Tuesday, she is once again awaken, interviewed and finally put back to sleep.
    In essence, the awakenings on Mondays and Tuesdays are indistinguishable to her.
    The most important question she's asked in the interviews is
    What is your credence (degree of belief) that the coin landed in heads?
    Given that Sleeping Beauty is epistemologically rational and is aware of all the rules of the experiment on Sunday, what should be her answer?
11. Alice secretly picks two different real numbers by an unknown process and puts them in two (abstract) envelopes. Bob chooses one of the two envelopes randomly (with a fair coin toss), and shows you the number in that envelope. You must now guess whether the number in the other, closed envelope is larger or smaller than the one you’ve seen.
    Is there a strategy which gives you a better than 50% chance of guessing correctly, no matter what procedure Alice used to pick her numbers?
12. Penny's game: Two player A & B choose a sequence of three coin tosses for eg. THT and HHT. Player A wins if we get THT before we get HHT after successive tosses of the coin . What sequence should you use to increase your chances of winning ?
13. There is a wolf at one vertex of a regular n-gon. There is a sheep at every remaining vertex. Each step, the wolf moves to a randomly chosen adjacent vertex and if there is a sheep, the wolf eat it. The wolf ends when it eats n-2 sheep (so there remains just one sheep) .Which sheep has maximum chance of survival?
14. What is the probability that two numbers randomly chosen are coprime?
15. If you are a pretty good basketball player, and were betting on whether you could make 2 out of 4 or 3 out of 6 baskets, which would you take?